Generative Adverserial Networks

Preliminaries.

A new framework for estimating generative models by training a generative network that captures the distribution of the data and a discriminative network which classifies whether the output of the generator is to have come from the training data itself.

The goal of the generator is to maximize the probability of the discriminator making a mistake, essentially generator $G$ and discriminator $D$ play a minimax game where one attempts to maximize for the objective one is trained to minimize.

Adverserial Nets.

The generator $G$ takes in a random noise variable $z$.

The discriminator $D$ takes in $x$, a vector of data (real or synthetic). $D$ is trained to maximize the probability of the output of $G$ being drawn / generated from the training data $\in x$.

$G$ is trained to minimze $\log\left(1 - D(G(z)) \right)$ such that the outputs of $D(\cdot)$ classify that $G(\cdot)$ correctly generated synthetic data from the real data.

The entire objective function is constructed as:

$$\min_G \max_D V(D, G) = \mathbb{E}_{x \sim p_{\text{data}}(x)}\left[\log D(x) \right] + \mathbb{E}_{z \sim p_x(z)}\left[\log(1 - D(G(z)))\right]$$

Note that $V(D, G) \in [-\infty, 0]$

where we want to find the $G$ which minimizes the value function $V$ but find the $D$ which maximizes $V$.

We're essentially playing a mini-max game, where $D$ is trained to maximize the value $V$, which essentially translates to $D$ being an optimal binary classifier between synthetic data non-synthetic data.

This is as if $D(x) = 1$, then $\log(D(x)) = 0$, then $\log(1 - D(G(z))) = \log(1) = 0$, and finally $\max V(\cdot) = 0$

When we attempt to find $G$ which satisfies $\min V(\cdot)$, we're try to find the $G$ which turns $D(G(x)) \rightarrow 1$ such that $\log(D(x)) + \log(\overline{.99}) \rightarrow -\infty$

Reminder that $D(x)$ is assumed to be a clasifier where $D(x) \rightarrow 1$ indicates that the input is real data while $D(x) \rightarrow 0$ indicates that the input is fake synthetic data

Then as we try to find the $G$ which satisfies $\max V(D, G)$ while we try to find the $D$ which satisfies $\min V(D, G)$, we're essentially playing what's called a zero-sum game where the gain in the generator is equal to the loss in the discrmininator (or vice versa).

Training.

For $k$ epochs and $m$ iterations = count of minibatches, update the discriminator by updating it's parameters, via gradient ascent,

$$\nabla_{\theta_d} V(D, G) = \frac{1}{m} \sum_{i = 1}^{m} \left[ \log D(x_i) + \log(1 - D(G(z_i))) \right]$$

while you update the generator by updating it's parameters via gradient descent,

$$\nabla_{\theta_g} V(D, G) = \frac{1}{m} \sum_{i = 1}^{m} \left[ \log(1 - D(G(z_i))) \right]$$

all via the update rule,

$$\theta^{(q)}_i = \theta_i^{(q)} ± \alpha \cdot \nabla_{\theta^{(q)}} V(D, G)$$

where $q \in g, d$ and $+$ if $q = d$, else $-$ if $q = d$.

The GAN is optimally trained when the discriminator can't distinguish the real data from the synthetic data -- this happens as we're trianing $G$ via gradient ascent which "confuses" or "fools" the discriminator from being able to optimally distinguish from both.